Each new state in FA3 is a combination of states from FA1 and FA2 in the following form eg.
z1 = x1 or y1 where x1 and y1 are start states for FA1 and FA2 respectively.

then you have to trace an a from x1 to xsomething and then trace an a from y1 to ysomething. You will now have a state called xsomething or ysomething. If this state is not the same as z1 (ie x1 or y1) then it becomes a new state called z2 so
z2 = xsomething or ysomething

You can now say in your transition table that an a takes you from z1 to z2. You do exactly the same for input leter b.

you have to trace a b from x1 to xsomething and then trace a b from y1 to ysomething. You will now have a state called xsomething or ysomething. If this state is not the same as z1 (ie x1 or y1) or z2 then it becomes a new state called z3 so
z3 = xsomething or ysomething.

You can now say in your transition table that a b takes you from z1 to z3.

This whole process is then repeated for the next state ie. z2, you trace a and b through the xsomething and ysomething of z2 to either create a new state or check that the state doesn't already exist. Eg an a through state z2 might take you back to z2 in which case a new state is not created.

I've gone through two examples in the textbook - one of where the input runs on both FA's simultaneously and the other where it starts on FA1 then either goes across to FA2 or stays on FA1.
Thnis might be a stupid question but how do we know which one they're asking for?

I'm stuck on the algorithm to chnage the FA from one for R to one for R*. I understand the example but the algorithm is confusing me. On page 129 of the textbook, step 1, what do they mean by 'create a state for every subset of x's' - what is a subset?